Mathematics Atlas

How Proof Is Made
Sign In
Text size
100%
Theme

Switch Atlas

Theorem

Euler's Identity

OY-ler, rhymes with boiler, not YOO-ler; named for Leonhard Euler
Also Known As Euler's Equation
Analysis
Unit Circle with a Radius Point

Euler's identity is the equation e to the power i times pi, plus 1, equals 0, linking five fundamental constants of mathematics: e, i, pi, 1, and 0. It is considered an exemplar of mathematical beauty, showing a profound connection between the most fundamental numbers in mathematics. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts
Disputed
Proof Year
1748 1
The identity follows from Euler's formula, published in his 1748 Introductio in analysin infinitorum, though historical records suggest Euler may never have explicitly written down this exact identity in this form himself.
Statement
Raising the base of the natural logarithm, e, to the power of i times pi, and adding 1, gives 0. The equation uses addition, multiplication, and exponentiation exactly once each. 1
Classification
Statement Form
Identity or Equation 1
Connections

Associated With

e (Euler's Number), Concepts

e is one of the three constants the identity equates to -1, alongside i and pi.

Additional Source Euler's Identity (Wikipedia)lead section (REST API extract, fetched 2026-08-30)
i (The Imaginary Unit), Concepts

i is one of the three constants the identity equates to -1, alongside e and pi.

Additional Source Euler's Identity (Wikipedia)lead section (REST API extract, fetched 2026-08-30)
Pi, Concepts

Pi is one of the three constants the identity equates to -1, alongside e and i.

Additional Source Euler's Identity (Wikipedia)lead section (REST API extract, fetched 2026-08-30)

In Branch

Analysis, Branches of Mathematics
Source Euler's Identity (Wikipedia)

Named After

Leonhard Euler, Mathematicians

Derived from the theorem's own name (unambiguous possessive-token match to exactly one live mathematician entity, w-bfill-g5-0924 browse backfill)

Proved By

Leonhard Euler, Mathematicians

Why this is disputed. Euler may never have explicitly written down this exact identity in this form, though it follows directly from his 1748 formula.

Source Euler's Identity (Wikipedia)
Sources
1. Euler's Identity (Wikipedia)
Wikimedia Foundation
  • lead paragraph
    Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation
  • history and attribution section
    historical records suggest Euler may never have explicitly written down this particular form
  • Associated With: e (Euler's Number), lead section (REST API extract, fetched 2026-08-30)
    Euler's number, the base of natural logarithms
  • Associated With: i (The Imaginary Unit), lead section (REST API extract, fetched 2026-08-30)
    is the imaginary unit, which by definition satisfies
  • Associated With: Pi, lead section (REST API extract, fetched 2026-08-30)
    is pi, the ratio of the circumference of a circle to its diameter.
View the Source
Euler (Wiktionary)
Wikimedia FoundationPronunciation section, English
Quote, Pronunciation section, English
/ˈɔɪlə(ɹ)/, (sometimes proscribed) /ˈjuːlə(ɹ)/
View the Source
Dissenting Readings (1 dissenting reading)
Proved By: Leonhard Euler

Euler's 1748 formula, e raised to the power i theta equals cosine theta plus i sine theta, implies the identity at theta equals pi, but no surviving source shows Euler himself explicitly writing the compact equation e to the i pi plus 1 equals 0. The identity's attribution to Euler by eponym reflects that it follows directly from his formula, not a documented instance of Euler writing this exact equation himself.

A dissenting reading, from a reviewerEuler's Identity (Wikipedia), Wikimedia Foundation
Comments (0)
No comments yet. Be the first to share a thought.
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.